Question: Let $S$ be a surface in 3D described by the equation $z = \sin(xy)$. Fill in the rest of the equation of the plane tangent to $S$ at $(0, \pi)$. $z = $
Solution: The equation for a tangent plane of an explicitly defined surface $z = f(x, y)$ at the point $(a, b)$ is: $f(a, b) + f_x(x - a) + f_y(y - b) = z$ [What's the intuition behind the formula?] We can see from the formula that the two values we're missing are $f(0, \pi)$ and $f_y$. $\begin{aligned} &f(0, \pi) = \sin(0\pi) = 0 \\ \\ &f_y = x\cos(xy) = 0\cos(0) = 0 \end{aligned}$ Here's the completed equation for the tangent plane of $S$ at $(0, \pi)$ : $z = 0 + \pi(x - 0) + 0(y - \pi)$